The Golden-Thompson inequality --- historical aspects and random matrix applications

TL;DR

This paper reviews the historical development and diverse applications of the Golden-Thompson inequality, especially in random matrix theory, highlighting new proofs, geometric interpretations, and norm generalizations.

Contribution

It provides a comprehensive survey including an unpublished proof, historical context, geometric insights, and applications to concentration inequalities in random matrices.

Findings

Unpublished Dyson proof of the inequality

Relation of the 2x2 case to hyperbolic geometry

Generalization to unitarily invariant norms

Abstract

The Golden-Thompson inequality, ${\rm Tr} \, (e^{A + B}) \le {\rm Tr} \, (e^A e^B)$ for $A,B$ Hermitian matrices, appeared in independent works by Golden and Thompson published in 1965. Both of these were motivated by considerations in statistical mechanics. In recent years the Golden-Thompson inequality has found applications to random matrix theory. In this survey article we detail some historical aspects relating to Thompson's work, giving in particular an hitherto unpublished proof due to Dyson, and correspondence with P\'olya. We show too how the $2 \times 2$ case relates to hyperbolic geometry, and how the original inequality holds true with the trace operation replaced by any unitarily invariant norm. In relation to the random matrix applications, we review its use in the derivation of concentration type lemmas for sums of random matrices due to Ahlswede-Winter, and Oliveira, generalizing various classical results.